Question

Evaluate
${}^{21}{C}_{15}{-}^{10}{C}_4$

Answer 1

Given

${}^{21}{C}_{15}-{}^{10}{C}_4=\frac{21!}{15!(21-15)!}-\frac{10!}{4!(10-4)!}$

$=\frac{21!}{15!6!}-\frac{10!}{4!6!}$

$=\frac{21\times 20\times 19\times 18\times 17\times 16\times 15!}{15!6!}-\frac{10\times 9\times 8\times 6!}{4!6!}$

$=\frac{21\times 20\times 19\times 18\times 17\times 16}{6!}-\frac{10\times 9\times 8\times 7}{4!}$

$=\frac{39070080}{720}-\frac{5040}{24}$

$=54264-210$

$=54054$